PHYSICAL REVIEW E 93, 013306 (2016)

Wideband analytical equivalent circuit for one-dimensional periodic stacked arrays

Carlos Molero,* Raul Rodrıguez-Berral,† and Francisco Mesa‡

Department of Applied Physics 1, University of Sevilla, 41012 Seville, Spain

Francisco Medina§

Department of Electronics and Electromagnetism, University of Sevilla, 41012 Seville, Spain

Alexander B. Yakovlev‖Department of Electrical Engineering, Center for Applied Electromagnetic Systems Research (CAESR),

University of Mississippi, University, Mississippi 38677, USA(Received 14 July 2015; published 15 January 2016)

A wideband equivalent circuit is proposed for the accurate analysis of scattering from a set of stacked slitgratings illuminated by a plane wave with transverse magnetic or electric polarization that impinges normally orobliquely along one of the principal planes of the structure. The slit gratings are printed on dielectric slabs ofarbitrary thickness, including the case of closely spaced gratings that interact by higher-order modes. A �-circuittopology is obtained for a pair of coupled arrays, with fully analytical expressions for all the circuit elements.This equivalent � circuit is employed as the basis to derive the equivalent circuit of finite stacks with any givennumber of gratings. Analytical expressions for the Brillouin diagram and the Bloch impedance are also obtainedfor infinite periodic stacks.

DOI: 10.1103/PhysRevE.93.013306

I. INTRODUCTION

The features of the scattering of electromagnetic waves byperiodic structures made of dielectric or metallic materialshave been studied for a long time at frequencies rangingfrom the optical [1–5] to the microwave [6–9] regime. Theinterest in these structures lies in their ability to controlthe polarization, reflection, transmission, and absorption ofelectromagnetic waves. Although fully dielectric implementa-tions are advisable at optical frequencies [10] and feasibleat microwave frequencies [11], many of the applicationsfound in the range from microwave to terahertz frequenciesinvolve metallic patterns printed on dielectric slabs. Thesepatterns are periodic distributions of strips or patches or,alternatively, periodic distributions of slits or slots etched onmetal films coating dielectric slabs. Structures of this kind havebeen commonly used as frequency-selective surfaces [12,13],polarizers [14,15], or high-impedance surfaces [16,17], tomention a few relevant examples.

The interest in these periodic structures was likewiseraised by the discovery of the so-called optical extraordinarytransmission [18], which was first associated with the char-acteristics of wave propagation at optical frequencies [18,19],although soon it was found and measured at millimeter wavebands [20,21]. In parallel with the previous works, someresearchers have proposed an alternative point of view basedon classical concepts of microwave circuit theory [6–9,22,23].This point of view was also implicitly adopted in some earlypapers on extraordinary transmission [24,25] and explicitly

*cmolero@us.es†rrberral@us.es‡mesa@us.es§medina@us.es‖yakovlev@olemiss.edu

employed in studies dealing with enhanced transmissionthrough small diaphragms inside closed waveguides [26–28].The possibilities of this perspective to provide a simpleand accurate explanation of the extraordinary transmissionbehavior and other exotic effects was initially employed bysome of the authors in papers that reported equivalent-circuitmodels to accurately obtain the transmission properties ofperiodic arrays of holes [29] or slits [30] made in thin orthick metal screens.

In most circuit-model-oriented papers, after the unit cell ofthe periodic structure is defined, the problem is reduced to thesolution of a typical waveguide-discontinuity problem, which,in general, can be substituted by an equivalent-circuit networkthat accurately reproduces the transmission and reflectionbehavior of the original structure. Moreover, the correspondingnetwork topology provides good physical insight into theinvolved electromagnetic phenomena and helps to predictother potentially interesting behaviors of the structure whenany structural parameter is varied. Another apparent advantageof the analytical equivalent-circuit approach is the short CPUtime required to achieve numerical results (usually in therange of milliseconds), in contrast to the great computationaleffort demanded by commercial electromagnetic full-wavesimulators (ranging from several minutes to hours). In theequivalent-circuit approach, the propagating fields are mod-eled by transmission lines, while the reactive fields aroundthe discontinuities are accounted for by lumped capacitanceor inductance elements, which are a good approximationfor electrically short discontinuity elements at the operatingwavelength [6,8]. Many equivalent-circuit proposals found inthe literature do not give analytical expressions for the lumpedelements, but instead their values are usually extracted fromintensive external full-wave simulations. In other instances[17,22,31–34] some analytical formulas are provided, butthey have a limited range of applications that typicallydoes not extend beyond the onset of the first grating lobe

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CARLOS MOLERO et al. PHYSICAL REVIEW E 93, 013306 (2016)

(diffraction regime) or are valid only in the subwavelengthregime. Further extension into the diffraction regime requiresmore sophisticated models, usually requiring that some ofthe circuit parameters are frequency dependent. In this way,some proposals [35–39] account for the diffraction ordersas different modes of the equivalent waveguide, whosemodal impedances provide the explicit closed-form frequencydependence of such circuit parameters. A further step wasgiven in [40], where, using a multimodal analysis, fullyanalytical expressions for all the circuit elements are provided(both lumped and distributed). In that work, self-consistentmodels for single metallic one-dimensional (1D) strip or slitarrays printed on dielectric layers were obtained by a rigorousmethod, and these models were extended to other periodicconfigurations such as T-shaped 1D periodic structures [41]and 2D arbitrarily shaped periodic structures [42]. Usinga similar multimodal analysis, diffractive effects were alsoconsidered in [38] for periodic distributions of slits madein thick metal screens. Another type of multimodal circuitmodels was derived, for instance, in [43]. A nice example ofhow circuit models can predict interesting physical effects isgiven in [44], where the anomalous extraordinary transmissionis explained as a consequence of the behavior of the circuitmodel associated with a 1D grating excited by transverseelectric (TE)-polarized waves. It should be mentioned here,to give proper credit to alternative paths to obtain highlyefficient solutions for this kind of problem, that analytical orquasianalytical solutions to the strip-like scattering problemare available not just in the form of equivalent circuits. Anumber of elegant and sophisticated mathematical tools basedon Wiener-Hopf and Riemann-Hilbert techniques as well asa variety of regularization methods to solve singular integralequations have been proposed, leading to extremely fast andaccurate computer codes (a few examples of this type of workcan be found in [45]–[53]). Equivalent circuits are usually lessaccurate but they have the advantage of being significantlysimpler than the above-mentioned approaches.

In all the above-mentioned models, the structure underanalysis involved a single grating surface, leaving asidethose situations where coupled gratings are of great interest[7,9,25,54–59]. The latter structures have been proposedfor design of artificial materials, filters, and polarizers, andvery recently they have been employed to simulate materialswith a high permittivity (artificial dielectrics) over a wideterahertz frequency range [60,61]. Some of the authors of thepresent work have already dealt [40,57] with a single pair ofcoupled identical strip-like gratings by using an even or oddexcitation analysis [62] (this allows us to reduce the originalcoupled-array problem to two subproblems with a singlegrating surface and with an electric or magnetic wall in thecorresponding symmetry plane), and a strategy similar to theone followed in this paper has very recently been implementedto approximately model fishnet structures [63]. Experimentalstudies of this type of structure can also be found for other 1D[54] and 2D [64] configurations.

As the obtaining of analytical solutions for stacked gratingsis a subject of considerable interest, the purpose of this workis first to present a systematic method to derive a widebandequivalent � circuit for a pair of 1D slit coupled arrays suchas that shown in Fig. 1 under normal or oblique transverse

FIG. 1. Cross section of a pair of periodic coupled slit arraysunder TM or TE normal incidence. The metal strips are infinitelythin and infinitely long along the x direction. Parameters: period ofthe arrays, p; slit width, w; distance between gratings (or dielectricthickness), d1; relative permittivity of the dielectric slab, ε(1)

r ; andrelative permittivity of the external medium, ε(0)

r .

magnetic (TM) or transverse electric (TE) incidence and thento use this � circuit as the basis to obtain the equivalentcircuit for the stacked structure. Fully analytical expressionsfor all the circuit elements are rigorously derived followingthe guidelines of our previous work on this topic [40,57]. Thiscircuit model can also be used to study infinite periodic stacksof 1D gratings. The dispersion relation of a Bloch mode andthe Bloch impedance of the structure is easily calculated fromthe unit-cell equivalent circuit.

II. CIRCUIT MODEL FOR TWO COUPLED SLIT ARRAYSUNDER NORMAL INCIDENCE

In this section the equivalent � circuit for a pair ofidentical periodic 1D coupled slit gratings is derived. Thestarting point is given in [40], where fully analytical circuitmodels were derived for single slit gratings with an electric ormagnetic wall placed at a certain distance from the slit array.These problems actually correspond to the odd-even excitationhalf-problems [62] of the symmetrical coupled grating. Thisfact is illustrated in Fig. 2, where the rightmost drawingshows a transmission line (representing the fundamental modepropagating in the external region) loaded with appropriateequivalent admittances. These admittances correspond to theinput admittances of the slit array printed on a dielectric slabof thickness d1/2 terminated with a magnetic (open-circuit;Y e

eq) or an electric (short-circuit; Y oeq) wall. The superscripts

“e” and “o” stand for even and odd excitations, respectively.

FIG. 2. Even-odd excitation analysis using magnetic or electricwalls in the symmetry plane and its circuit representation accordingto Ref. [40]. The superscript “e” stands for even excitation; thesuperscript “o,” for odd excitation.

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According to [40], Y eeq and Y o

eq are given by the infinite sum ofmodal admittances,

Y eeq =

∞∑n=1

AnY(0)n + j

∞∑n=0

AnY(1)n tan

(β(1)

n d1/2), (1)

Y oeq =

∞∑n=1

AnY(0)n − j

∞∑n=0

AnY(1)n cot

(β(1)

n d1/2), (2)

where

An = (2 − δn0) ×{

J 20 (knw/2), TM incidence,[2 J1(knw/2)

knw/2

]2, TE incidence,

(3)

Y (i)n = 1

η0×{ε(i)

r k0/β(i)n , TM incidence,

β(i)n /k0, TE incidence,

(4)

β(i)n =

√ε

(i)r k2

0 − k2n, (5)

k0 = ω

c, kn = 2πn

p, (6)

where δij is the Kronecker delta, Ji(·) the Bessel function of thefirst kind and ith order, η0 = √

μ0/ε0 the intrinsic impedanceof free space, ω the angular frequency, c the speed of lightin free space, kn the cutoff wave number of the nth modesupported by the parallel-plate waveguide associated with theunit cell, and β(i)

n the corresponding longitudinal wave numberof the nth mode in medium (i). The An coefficient is thenth Fourier coefficient of the normalized electric-field profileassumed at the slit aperture (y ∈ [−w/2,w/2], with w beingthe width of the slit), which is taken as

Eslit(y) ={

2πw

[1 − (2y/w)2]−1/2, TM incidence,4

πw[1 − (2y/w)2]1/2, TE incidence

(7)

(note that this approximation accounts for the correct edgebehavior of the aperture field in both the TM and the TEcases).

Starting from the basic circuit models for the even- andodd-excitation-half problems shown in Fig. 2, our first goal inSec. II A is to find an equivalent circuit for the pair of coupledgratings in the form of an equivalent � network. Then Sec. II Bpresents a detailed study of the elements in the � circuit,leading to a further decomposition into different elementsthat allows for their efficient computation and provides aninsightful physical interpretation. Finally, Sec. II C presentsthe strategy proposed for the computation of the �-circuitelements according to the discussion in Sec. II B, and Sec. II Dpresents various numerical results that serve to illustrate andsupport the previous discussion as well as to demonstrate thequantitative performance of the �-network circuit model.

A. Derivation of the equivalent � network for coupled gratings

In order to find the equivalent � network that describesthe behavior of the coupled screens [see Fig. 3(a)], an even-odd excitation analysis is also applied to this � circuit byplacing open- or short-circuit terminations in the middle plane[Fig. 3(b)]. As a result, the half-problem circuits shown inFigs. 3(c) and 3(d) are obtained. Identifying the latter two

(a)

(b)

(c) (d)

FIG. 3. (a) Equivalent � circuit for a pair of coupled gratings.(b) The same circuit as in (a), showing the middle symmetry plane.(c, d) The circuits that result from applying open or short circuitterminations (corresponding to even or odd excitations) at the middlesymmetry plane.

circuits with the one in Fig. 2, it is clear that the parallel andseries admittances (Yp and Ys) can readily be obtained fromthe even and odd equivalent admittances as

Yp = Y eeq, Ys = 1

2

[Y o

eq − Y eeq

], (8)

and therefore, from (1) and (2),

Yp =∞∑

n=1

AnY(0)n + j

∞∑n=0

AnY(1)n tan

(β(1)

n d1/2), (9)

Ys = −j∞∑

n=0

AnY(1)n csc

(β(1)

n d1). (10)

It should be noted [57] that the first summation in Yp,

Y (0)p =

∞∑n=1

AnY(0)n , (11)

only accounts for the external field, whereas the secondsummation is associated with the field in region (1):

Y (1)p = j

∞∑n=0

AnY(1)n tan

(β(1)

n d1/2). (12)

This fact is key in Sec. III to obtaining the equivalent circuit forstacked structures involving an arbitrary number of gratings.

B. Interpretation of the admittances deduced for the � circuit

The analytical expressions found in previous sections forthe circuit elements of the � network have been derived asinfinite series, and certainly, these formal expressions are notwell suited for a qualitative understanding of the behaviorof the fields, nor are they convenient from a computationalperspective. However, it will be shown that they can be easilymanipulated and rewritten in a more insightful and suitableform. First, it is convenient [40,43] to split the infinite series in

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(10), (11), and (12) into two separate contributions. The firstcontribution comprises the low-order terms, which correspondto propagating modes at the highest frequency of interest. It isalso advisable to add the first evanescent mode to this low-ordergroup. The second contribution is formed by an infinite seriesof evanescent modes whose cutoff frequencies are far abovethe working frequency, namely, high-order (“ho”) evanescentmodes. Expressions in (10), (11), and (12) are thus regroupedas

Y (0)p =

N∑n=1

AnY(0)n +

∞∑n=N+1

AnY(0)n︸ ︷︷ ︸

Y(0)p,ho

, (13)

Y (1)p = j

N∑n=0

AnY(1)n tan

(β(1)

n d1/2)

+ j∞∑

n=N+1

AnY(1)n tan

(β(1)

n d1/2)

︸ ︷︷ ︸Y

(1)p, ho

, (14)

Ys = − jN∑

n=0

AnY(1)n csc

(β(1)

n d1)

− j∞∑

n=N+1

AnY(1)n csc

(β(1)

n d1)

︸ ︷︷ ︸Ys, ho

, (15)

where N is the number of low-order (“lo”) terms. Accordingto the above-mentioned criterion, N will be given by

N =⌈√

εr,maxp

λ0,min

⌉, (16)

where �·� represents the smallest following integer (ceilingfunction), εr,max is the greater of the two surrounding permit-tivities, and λ0,min is the minimum vacuum wavelength valueof interest. Since the modes that contribute to the Y

(i)p,ho and Ys,ho

high-order admittances are far below their cutoff frequencies,the following approximation can be used:

kn �√

ε(i)r k0, β

(i)n,ho ≈ −jkn (i = 0,1). (17)

The expressions for the TM and TE characteristic admittancesof these modes then reduce to

Y(i)n,ho ≈ jω

ε0ε(i)r

kn

= jωC(i)n , TM case, (18)

Y(i)n,ho ≈ 1

jω

kn

μ0= 1

jωLn

, TE case, (19)

whose frequency dependence is readily recognized as standardreactive lumped elements (inductors and capacitors). Depend-ing on the polarization of the impinging wave, a lumpedn-order capacitor C(i)

n or n-order inductance Ln is definedfor the nth-order mode (note that Ln does not depend onthe permittivity of the medium, as expected for a standardinductance).

For TM polarization, introducing (18) in Y (i)p and Ys, we

obtain

Y(0)p, ho = jω

∞∑n=N+1

AnC(0)n = jωC(0)

p , (20)

Y(1)p, ho = jω

∞∑n=N+1

AnC(1)n tanh(knd1/2) = jωC(1)

p , (21)

Ys, ho = jω∞∑

n=N+1

AnC(1)n csch(knd1) = jωCs, (22)

where C(1)p and Cs are capacitors that account for the global

effect of all the higher-order evanescent modes.Similarly, for TE polarization the following equivalent

inductances associated with the effect of all the higher-ordermodes are obtained:

Y(0)p, ho = 1

jω

∞∑n=N+1

An/Ln = 1

jωL(0)p

, (23)

Y(1)p, ho = 1

jω

∞∑n=N+1

(An/Ln) tanh(knd1/2) = 1

jωL(1)p

, (24)

Ys, ho = 1

jω

∞∑n=N+1

(An/Ln) csch(knd1) = 1

jωLs. (25)

The circuit models in the form obtained thus far are depictedin Fig. 4, with the high-order capacitances and inductancesexplicitly shown. For simplicity, in these drawings it isassumed that N = 0; i.e., the frequently encountered situationin which only the fundamental zero-order term contributesto the low-order admittances that appear in the internal �

network. All the higher-order modes are then included in thelumped elements.

As pointed out for the single-grating problem [40], amore specific physical interpretation of these lumped reactiveelements can be found after noting that they represent the

(a)

(b)

FIG. 4. (a) Equivalent circuit showing separately the contributionof the fundamental mode (� network) and the higher-order modes(lumped capacitors and inductors). (a) TM polarization; (b) TEpolarization.

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WIDEBAND ANALYTICAL EQUIVALENT CIRCUIT FOR . . . PHYSICAL REVIEW E 93, 013306 (2016)

reactive field excited in the vicinity of the slit aperture (sincethey account for the evanescent modes). This interpretationcan directly be translated to the external high-order element(C(0)

p or L(0)p ), which represents the reactive near field around

the slit in the external region. For the internal region it is alsodesirable to find an interpretation for both the series and theparallel reactive lumped elements. For the series element in(22) or (25), the exponential behavior of the csch(·) functionfor large arguments makes this series admittance decreaseexponentially as the two coupled screens are separated, whichclearly suggests that the series element is basically related tothe coupling between both screens through evanescent fields.From (22) or (25) it is apparent that only the evanescentmodes that reach the other screen will contribute significantlyto the series element, with the contribution of the remainingmodes being negligible. On the contrary, the parallel elementin (21) or (24) does not vanish for distant screens, sincethe tanh(·) function approaches unity instead. Therefore, forcoupled screens sufficiently separated (so that the reactivefield around the slits does not reach the other screen), theparallel lumped element approaches its single-grating value.This suggests further splitting the parallel element as

C(1)p =

∞∑n=N+1

AnC(1)n︸ ︷︷ ︸

C(1)p, single

−∞∑

n=N+1

AnC(1)n [1 − tanh(knd1/2)]

︸ ︷︷ ︸Cp, coup

.

(26)The first term, C

(1)p, single, represents the value of the reactive

element in the absence of coupling with a second screen(single-grating value). The second one, Cp,coup, is a couplingterm that accounts for the modification of the single-screencapacitance due to the presence of the other screen. Thefact that this term is subtracted from C

(1)p, single indicates

that the overall parallel capacitance tends to decrease as aconsequence of the coupling, and this can be interpreted as aparallel connection between the single-screen capacitance anda negative capacitance, −Cp,coup. A similar decomposition canbe made for the parallel inductance in the TE case as

1/L(1)p =

∞∑n=N+1

An/Ln︸ ︷︷ ︸1/Lp, single=1/L

(0)p

−∞∑

n=N+1

An/Ln[1 − tanh(knd1/2)]

︸ ︷︷ ︸Lp, coup

.

(27)As found for the series coupling elements in (22) and (25),only those higher-order modes whose evanescent field reachesthe other screen need to be included in the parallel couplingelements (26) and (27). A convenient criterion to determinethe highest-order mode, M , to be considered in the Cs, Ls,Cp,coup, and Lp,coup summations in (22), (25), (26), and (27) isfound to be

kMd1 ≈ 1 =⇒ M =⌈

1

2π

p

d1

⌉. (28)

This criterion has been satisfactorily checked in situations inwhich coupling through evanescent modes play a significantrole.

Finally, in Fig. 4 the contribution of the zero-order modeis accounted for by the internal � circuit formed by the

FIG. 5. Circuit models for (a) TM and (b) TE polarizationsconsidering the fundamental mode as the single distributed element(transmission line of admittance Y

(1)0 ). The higher-order-mode con-

tribution is included in the lumped elements. (c) Circuit model forthe case of two distributed modes (only the TM polarization case).

Y(1)p, 0 and Ys,0 admittances. After some manipulations (see the

Appendix), it can be shown that such a low-order � circuitis equivalent to a transmission-line section with characteristicadmittance Y

(1)0 and length d1.

The above ideas are illustrated in the circuits shownin Fig. 5, with the aim of gaining a graphical insightfulinterpretation (it is again assumed that the zero-order modeis the only low-order term; N = 0). The internal transmissionline with admittance Y

(1)0 accounts for the coupling between

the screens through the propagating zero-order wave. Thereactive elements labeled “single” represent the reactive fieldaround the slits for a single grating (i.e., in the absenceof coupling through evanescent fields). For TM polarizationthis element is obtained as the sum (parallel connection) ofC(0)

p in (20) and C(1)p,single in (26) (for TE polarization it is

given by the parallel connection between the correspondinginductances). If the two coupled screens are distant enoughso that there is not significant coupling through evanescentmodes (namely, if M = 0 for the cases in Figs. 5(a) and 5(b)

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CARLOS MOLERO et al. PHYSICAL REVIEW E 93, 013306 (2016)

or M � N in general), then the reactive coupling elementscan be neglected. This situation, with only one propagatingmode and no reactive coupling, is obtained when the period iselectrically small and the dielectric slab is electrically thick. Inthe literature, many authors restrict themselves to the analysisof such a situation. However, closely spaced gratings areimportant for many practical cases of interest, and in suchcases, the coupling through evanescent fields should actuallybe taken into account, which can conveniently be carried outby introducing the series and parallel coupling capacitorsor inductors. Furthermore, in order to extend the range ofapplications of the model to higher frequencies where oneor more higher-order modes become propagating in the slabbetween the gratings, the full dynamic frequency dependenceof their modal admittances and propagation wave numbers hasto be considered. As N > 0, these higher-order modes mustcontribute to the low-order sums in (13), (14), and (15) and itsrespective effect can similarly be separated from the high-orderelements to form another � circuit, which is again equivalentto another transmission line with its corresponding admittanceY

(1)i and length d1. This fact is sketched in Fig. 5(c) for N = 1

and TM polarization. The appearance of transformers of turnratio A

1/21 placed at both ends of the transmission line is

noteworthy.

C. Strategy for the computation of the �-circuit admittances

Taking into account the ideas discussed above, we nowdiscuss the systematic strategy followed in this work tocompute the Y (0)

p , Y (1)p , and Ys admittances that make up the

� circuit. First, given the structural parameters and frequencyrange of interest, the value of N is computed following thecriterion proposed in (16). Using this value of N , the C(0)

p and

C(1)p,single lumped elements in (20) and (26) are computed as

C(0)p = wε0ε

(0)r

∞∑n=N+1

[J0(knw/2)]2

knw/2, (29)

C(1)p,single = ε(1)

r

ε(0)r

C(0)p . (30)

Similarly, for the TE case, the L(0)p and L

(1)p,single elements in

(23) and (27) are obtained from

1

L(0)p

= 1

L(1)p,single

= 16

wμ0

∞∑n=N+1

[J1(knw/2)]2

knw/2. (31)

Next, the value of M is obtained from the criterion in (28). IfM > N , then the coupling reactive elements are computed as

Cp,coup = wε0ε(1)r

M∑n=N+1

[J0(knw/2)]2

knw/2[1 − tanh(knd1/2)],

(32)

Cs = wε0ε(1)r

M∑n=N+1

[J0(knw/2)]2

knw/2csch(knd1) (33)

or

1

Lp,coup= 16

wμ0

M∑n=N+1

[J1(knw/2)]2

knw/2[1 − tanh(knd1/2)],

(34)

1

Ls= 16

wμ0

M∑n=N+1

[J1(knw/2)]2

knw/2csch(knd1). (35)

If M � N , these coupling elements are ignored. Once all of theabove lumped elements are known, the high-order admittancesin (13)–(15) are given by

Y(0)p,ho = jωC(0)

p , (36)

Y(1)p,ho = jω

(C

(1)p,single − Cp,coup

), (37)

Ys,ho = jωCs, (38)

and the corresponding expressions for the TE case are obtainedby replacing jωC with 1/jωL. It is important to note thatnone of the above lumped elements depends on frequency.Hence, they have to be evaluated only once when performinga frequency sweep. Furthermore, the elements with infinitesummations in (29)–(31) do not depend on d1 either, andtherefore it is not necessary to recalculate them when varyingthe distance between the gratings, provided the period p or theslit width w is not modified. If so desired, diverse techniquescan be applied to find closed-form expressions for the involvedseries [23]. More specifically, all the numerical series requiredin this paper, which involve Bessel functions in their generalterm, have been computed in closed form in the frame of theanalysis of the so-called boxed microstrip line [65,66]. Thus,the results in [65] and [66] can be directly employed to speedup the computation of the series in the present work. Finally,for each value of frequency, the admittances in the � circuitare obtained as

Y (0)p =

N∑n=1

AnY(0)n + Y

(0)p,ho, (39)

Y (1)p = j

N∑n=0

AnY(1)n tan

(β(1)

n d1/2)+ Y

(1)p, ho, (40)

Ys = −jN∑

n=0

AnY(1)n csc

(β(1)

n d1)+ Ys, ho. (41)

D. Results and discussion

Some numerical results are next presented in order to checkthe validity and accuracy of the equivalent-circuit approachfor a pair of coupled gratings and to illustrate the previousdiscussion. Figure 6 shows the transmission coefficient of apair of coupled gratings printed on either side of a dielectricslab, for two values of the slab thickness d1 (separationbetween the gratings).

For comparison purposes, we have developed an in-housenumerical code based on the method of moments (MoM)solution of the integral equation for the aperture field. ThisMoM implementation, which uses several basis functions to

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00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

|S21

|

Normalized frequency p/λ0

MoMN =0N =1N =2

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

|S21

|

Normalized frequency p/λ0

MoMN =0N =1N =2N =2, M =8

(a)

(b)

FIG. 6. Magnitude of the transmission coefficient (|S21|) for twocoupled gratings versus the normalized frequency for TM incidenceand for different thicknesses (d1) of the dielectric slab between thegratings (i.e., for different values of the separation between thegratings). Parameters: w = 0.1p, ε(0)

r = 1, ε(1)r = 4. (a) Relatively

distant gratings: d1 = p/2.5 = 4w. (b) Closely spaced gratings:d1 = p/50 = w/5.

reproduce the aperture field, has been carefully checked andthe numerical results it provides can be considered virtuallyexact for our purposes.

In the first case shown in Fig. 6(a), the separation betweenthe metallic screens is relatively large (the slab thickness is2.5 times smaller than the periodicity of the slit gratings).Indeed, the value of the ratio p/(2πd1) is quite low (≈0.4),and consequently, the coupling through evanescent fields is notexpected to be relevant. Hence, the Cp,coup and Cs capacitorsdo not have to be considered in the circuit model [in otherwords, M = 0 is taken for the circuit-model results shownin Fig. 6(a), although M = 1 could be taken according toour more conservative general criterion given in (28)]. It isshown in the figure that the circuit-model results obtainedwith N = 0 reproduce accurately the transmission spectrumat low frequencies, but they tend to deviate as the frequencyincreases until they become qualitatively wrong. The reason

is that the first higher-order mode becomes propagating in thedielectric slab at p/λ0 = 0.5. Thus, the N = 0 results startto deviate at lower frequencies (p/λ0 � 0.2) where the n = 1evanescent mode is no longer far below its cutoff frequency andits increasingly dynamic behavior starts to have an effect on thetransmission spectrum. The circuit model can accommodatethis situation simply by setting N = 1, i.e., by incorporatingthe corresponding � network for the n = 1 mode (togetherwith that for the fundamental n = 0 wave, which is alwayspresent). This is confirmed by the N = 1 results in Fig. 6(a),which show a very good agreement with the MoM curve up top/λ0 � 0.6. For higher frequencies, these results show somedeviation with respect to the MoM data, although they arestill able to reproduce the main features of the spectrum. Onceagain, this deviation is due to the fact that the next-higher-ordermode (n = 2) is approaching its cutoff frequency inside thedielectric slab (p/λ0 = 1). The circuit model with N = 2 doesprovide very accurate results over the whole frequency range.

Next, Fig. 6(b) shows the transmission spectrum for thesame two coupled slit gratings as in Fig. 6(a), but now witha much thinner dielectric slab (d1 = p/50), and hence, thegratings are tightly coupled. Two total transmission peaksare observed, each one followed by a transmission zero. Ifonly the zero-order wave is considered as distributed elementin the model through its corresponding � network (curvelabeled N = 0), the resulting equivalent circuit is not ableto reproduce the transmission behavior at all. When the firsthigher-order mode is considered a “low-order” mode and thusits dynamic distributed nature is taken into account through thecorresponding � network (N = 1 curve), the first transmissionpeak and zero are captured by the circuit model, though slightlyblue-shifted. The inclusion of the second higher-order mode(N = 2) mostly corrects this frequency shift and makes theproposed circuit model able to reproduce the second peakand zero. However, even with N = 2 the results are notsatisfactory, since a significant difference in the transmissionlevel with respect to MoM results is observed throughout theconsidered frequency range. The reason is that the circuit usedto compute the N = 0, N = 1, and N = 2 curves in Fig. 6(b)does not include the Cp,coup and Cs coupling capacitors, whichimplies that the coupling through evanescent modes is beingneglected. As the gratings are now very close to one another,this coupling is expected to be significant. Indeed, for thepresent configuration and according to our criterion in (28),M should be set to 8. This means that, for the circuit modelwith N = 2, higher-order evanescent modes from n = 3 ton = 8 should contribute to the Cp,coup and Cs capacitors. Whenthese capacitors are also introduced in the circuit model (curvelabeled N = 2, M = 8), the agreement with the MoM resultsis very good over the whole frequency range shown. At thispoint, it is also worth highlighting the computational efficiencyof the proposed equivalent-circuit method. Even in this quitestringent case (with three distributed modes and up to eightmodes contributing to the evanescent coupling), the CPU timeneeded for the computation of the ≈1000 values representedin Fig. 6(b) is almost negligible (it takes about 100 ms in amodest personal computer).

A second example is given in Fig. 7, which shows a compar-ison between our equivalent-circuit results and experimentalresults reported in Ref. [54]. The transmissivity coefficient

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CARLOS MOLERO et al. PHYSICAL REVIEW E 93, 013306 (2016)

FIG. 7. Comparison of our circuit approach data with experi-mental results [54] for the transmissivity coefficient (|S21|2) of apair of tightly coupled 1D gratings when a TM-polarized normallyincident plane wave impinges on the structure. Parameters: p =10.02 mm, w = 0.29 mm, d1 = 0.356 mm, ε(0)

r = 1, ε(1)r = 4.17,

tan(δ) = 0.0167.

is plotted versus the frequency when a normally incidentTM plane wave impinges on a pair of coupled arrays. Thegratings are printed on the two sides of an electrically verythin and lossy FR4 substrate. It should be pointed out thatlosses in the substrate can be straightforwardly incorporatedin the circuit model through the admittances, (4), and wavenumbers, (5), just by replacing the lossless real value ofpermittivity, ε(i)

r , with its complex value for lossy substrates,ε(i)

r (1 − j tan δ(i)). According to our general criteria in (16) and(28), the circuit-model results in Fig. 7 are computed withN = 2, M = 5. As can be observed, the agreement with theexperimental results is remarkably good.

The case of TE incidence is studied in Fig. 8, whichshows the frequency behavior of the transmission coefficientwhen a normally incident plane wave impinges on a pair

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

|S21

|

Normalized frequency p/λ0

MoMCircuit Model

FIG. 8. Transmission coefficient (magnitude) versus frequencyfor TE incidence. Parameters: w = 0.2 p , d1 = 0.5 p, ε(0)

r = 1, ε(1)r =

9.8.

of coupled gratings. For this polarization the interaction ofthe incident wave with the slits is weaker than for the TMcase and an electrically thick medium is needed to observetransmission peaks, which are extremely frequency selective.Four low-order terms were taken as distributed elements(N = 4) because of the high permittivity of the substrate.These terms are associated with propagating modes in thedielectric region since the interaction through evanescentmodes is almost negligible because the screens are far enoughto be affected by them (M = 0). Once again, the accuracyshown by the circuit model compared with the numerical MoMdata is very good.

Concerning the validity limits of the approach, it isthe assumption of a frequency-independent field profileat the slits, (7), that constitutes the fundamental limitationin the application of the model. Therefore, the model isvalid for relatively narrow slits; i.e., the results provided bythe equivalent circuit will start to deteriorate for frequenciesat which the slits are not electrically small. In order toqualitatively estimate the limits of validity, we have carried outextensive numerical experiments for different configurationsunder both polarizations. For TM polarization, the model isfound to provide reliable results for w/λdiel � 0.4, where λdiel

is the wavelength inside the highest-permittivity dielectricinvolved. Under TE polarization, the numerical experimentsshowed that the validity limit can be estimated as w/λeff �0.75, where λeff is the wavelength inside an effective mediumwith relative permittivity εeff = (ε(0)

r + ε(1)r )/2.

III. STACKED ARRAYS

The previous section presented the derivation of the �-circuit model for a pair of coupled slit gratings, together witha detailed in-depth physical interpretation of its elements.All the previous derivations and discussions will be foundto be very pertinent for the study of stacked arrays since itscorresponding equivalent circuit will be systematically builtup from the �-circuit model for a pair of coupled slit gratings.As already pointed out in Sec. II A, the key fact is that theYp admittance in the � network is given by the sum (parallelconnection) of the external, Y (0)

p , and internal, Y (1)p , parallel

admittances. Since these admittances independently accountfor the field in medium (0) (external medium) and medium(1) (internal medium), an “internal” � block associated withmedium (1), characterized by Y (1)

p and Ys, can be defined asshown in Fig. 9. This internal � circuit can be employedas a “building block” to build up the equivalent circuit of aseries of stacked arrays, as shown in Fig. 10. The stackedstructure amenable to analysis using this procedure may have

FIG. 9. Topology of the equivalent circuit defining the internal �

circuit (building block) and the external part.

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WIDEBAND ANALYTICAL EQUIVALENT CIRCUIT FOR . . . PHYSICAL REVIEW E 93, 013306 (2016)

impingingplane wave

slit gratings

(a)

(b)

...2dd1 d3 dN

FIG. 10. (a) Stacked structure with different dielectric slabs.(b) Equivalent circuit made by cascading � building blocks.

dielectric layers of different thickness and permittivity, butthe slit arrays must be identical and aligned among them topreserve the periodicity in the transverse direction. It should benoted that the employed strategy is applicable to finite stackedstructures thanks to the initial “aperture” formulation of theproblem, which made it possible to define two independentsubproblems at both sides of the apertures (slits).

The above rationale can also be employed to study aninfinite set of periodically stacked slit gratings. In this case,the equivalent network of this longitudinally periodic structureis formed by the periodic repetition of the corresponding� building block (the “unit cell” of the structure alongthe longitudinal z direction), as illustrated in Fig. 11. Asthe elements of this � building block are known in closedform, the dispersion relation of the Bloch modes of the infinite

(a)

(b)

FIG. 11. (a) Infinite periodic stack of slit gratings. (b) Equivalentcircuit consisting of an infinite cascade of � building blocks.

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

|S21

|

Normalized frequency p/λ0

Circuit ModelHFSS

FIG. 12. Magnitude of the transmission coefficient (|S21|) versusthe normalized frequency for four stacked slit arrays under normalTM incidence. Parameters: w = 0.1 p, d1 = 0.4 p, d2 = 0.3 p, d3 =0.2 p, ε(0)

r = 1, ε(1)r = 2.2, ε(2)

r = 4, ε(3)r = 3.

periodic stack is readily obtained as [62]

cosh(γ d1) = 1 + Yp

Ys, (42)

where γ = α + jβ is the complex propagation constant ofthe Bloch mode, with β being the phase constant and α theattenuation constant. The Bloch admittance at the longitudinalunit cell terminals is also derived immediately as [62]

YB = √Yp(Yp + 2Ys). (43)

A. Results and discussion

As a first example to check the validity of the equivalent-circuit approach for stacked arrays, the transmission coefficientof a set of four cascaded arrays printed on three dielectricslabs under normal TM incidence is plotted in Fig. 12. A highreflection behavior is observed at the center of the plot, from0.25p/λ0 up to about 0.7p/λ0. At low frequencies, a high levelof transmission is expected because the slit gratings interactweakly with the incident wave. Additional transmission peaksare also found between 0.7p/λ0 and 0.9p/λ0. For comparisonpurposes, the results provided by the commercial simulatorAnsys HFSS [67] (based on the finite-element method in thefrequency domain) are also shown. An excellent agreement isfound between the two sets of data.

Next, the case of an infinite periodic stack of identicalgratings is studied in Fig. 13. The Brillouin diagram for theBloch mode of this periodic structure obtained from (42) isplotted in Fig. 13(a), which shows four passbands separatedby three stopbands. It is interesting to note that the slope of thedispersion relation is positive in the first two passbands (oftencalled forward passbands), whereas it is negative in the twohigher-frequency bands (backward passbands). Since no lossesare assumed in any element of the present structure, the secondmember of (42) is always real, providing two possibilities for

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CARLOS MOLERO et al. PHYSICAL REVIEW E 93, 013306 (2016)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Nor

mal

ized

freq

uenc

yp/

λ 0

(a)

(b)

(c)

-400

-300

-200

-100

0

100

200

300

400

0 0.2

0.4 0.6

0.8 1

Blo

chIm

peda

nce

ZB

(Ohm

s)

0.4 0.6

Real part of ZBImaginary part of ZB

Normalized frequency p / λ0

FIG. 13. (a) Brillouin diagram of an infinite stacked structure.(b) Transmission coefficient (dB) versus normalized frequency fora truncated version (10 stacked gratings). (c) Bloch impedance ofthe Bloch mode represented in (a). The parameters of the unit cellare w = 0.15 p, d1 = 0.3 p , ε(0)

r = 1, ε(1)r = 4. Frequency regions

labeled SB 1, SB 2, and SB 3 denote stopbands.

the propagation constant:

γ = jβ if

[1 + Yp

Ys

]∈ [−1,1], (44)

γ = α + j

{0πd

if

[1 + Yp

Ys

]�∈ [−1,1]. (45)

Clearly, passbands occur when α = 0 (γ = jβ), and stopbandswhen α �= 0 and β = 0 or πd. The existence of a passband isthen subject to the following two conditions: (i) the reactivenature (capacitive/inductive) of Ys and Yp has to be different inorder to satisfy Yp/Ys < 0 , and (ii) |Yp| < 2|Ys|. For instance,these conditions are satisfied if the � network has a relativelysmall capacitance as a parallel element (close to an opencircuit) and a low inductance as a series element (close toa short circuit), which allows the power to pass through thestructure, thus obtaining a forward band [62]. On the otherhand, the above two conditions can also be satisfied if Ys iscapacitive and Yp inductive; for instance, a � network with ahigh capacitance as a series element and a high inductance asthe parallel element. This would also allow the power to travelalong the stacked structure, but now the passband would be ofa backward nature.

If a finite (truncated) version of the previous longitudinallyperiodic structure is now considered, a clear correspondenceis expected between the Brillouin diagram and the behaviorof the transmission coefficient of the finite stack. This factis illustrated in Fig. 13(b), which shows the transmissioncoefficient through a finite stack with 10 gratings. The presenceof successive passbands or stopbands that perfectly correlatewith the associated Brillouin diagram in Fig. 13(a) can beobserved. The circuit-model results (solid lines) are comparedwith those provided by the Ansys HFSS (circles), showingan excellent agreement. Another feature of this transmissionbehavior worthy of mention is the high level of ripple observedin the passbands. This is due to impedance mismatchingbetween the impinging wave (whose characteristic impedanceis the intrinsic impedance of the free space, η0 ≈ 377 �) andthe Bloch mode of the stack. Figure 13(c) shows the realand imaginary parts of the Bloch impedance (ZB = 1/YB)obtained from (43). Since the considered structure is lossless,ZB is real-valued at passband and purely imaginary at stopbandfrequencies. For the most part of the first three passbands,the ZB value is much lower than the free-space impedance,although it tends to grow within each band, which is consistentwith the decreasing trend of the ripples. Also, ZB is higher inthe first (low-frequency) passband, and hence the ripples in thisband are smaller than in the next two passbands. Finally, thelast passband (at the higher frequencies) is the one showing byfar the highest mismatching, due to the particularly low valueof ZB at these frequencies.

As the last example, in Fig. 14(a) a lossy stack with thespace between the gratings filled with resistive silicon slabsis studied. The complex effective permittivity of the slabs isgiven by ε(1) = ε0ε

(1)r − jσ/ω, where σ is the dc conductivity

of the layers. Figure 14(a) shows the power absorbed in afinite stack (normalized to the incident power), whereas theBloch impedance of the corresponding infinite stack is plottedin Fig. 14(b). Since the structure is lossy, the Bloch impedancetakes complex values at any frequency. The most interestingfeature is the high-absorption band at high frequencies, witha fractional bandwidth of 4.5% for a central frequency of22.5 GHz. As shown in Fig. 14(b), this absorption band canbe related to a Bloch impedance with a relatively constantreal part that is close to the free-space impedance and with arelatively small imaginary part in the same frequency range.This behavior of the Bloch impedance has been obtained by

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WIDEBAND ANALYTICAL EQUIVALENT CIRCUIT FOR . . . PHYSICAL REVIEW E 93, 013306 (2016)

(a)

(b)

FIG. 14. (a) Power absorbed in a lossy system formed by eightstacked arrays separated by seven resistive (ohmic) dielectric slabs(σ = 0.2 S/m). (b) Bloch impedance versus frequency. Parameters:p = 5 mm, w = 1.4 mm, d1 = 2.2 mm, ε(0)

r = 1, ε(1)r = 11.9.

properly adjusting the geometrical parameters and the valueof the slab conductivity. Note that, for frequencies betweenapproximately 17 and 20 GHz, ZB is mostly real-valued aswell. However, its small value implies a strong mismatch withthe incident wave that causes most of the incident power to bereflected rather than absorbed in the resistive silicon slabs.

IV. CIRCUIT MODEL FOR OBLIQUE INCIDENCE

The extension of the circuit model to the case of a planewave that impinges obliquely with an incident angle θ isstraightforward. Assuming that the incidence is in the yz plane(see Fig. 1), the corresponding incident wave vector is givenby

k =√

ε(0)r k0(cos θ z + sin θ y). (46)

The parallel and series admittances obtained in Sec. II aredefined in terms of an infinite sum of modes. For oblique

incidence, the transverse boundary conditions of the unit cellare no longer electric or magnetic walls but periodic walls,which implies that the series of modes now become seriesof Floquet spatial harmonics. Thus, the series and paralleladmittances in Fig. 9 can be written as

Y (0)p =

∞∑n = −∞n �= 0

AnY(0)n , (47)

Y (1)p =

∞∑n=−∞

AnY(1)n tan

(β(1)

n d1/2), (48)

Ys =∞∑

n=−∞AnY

(1)n csc

(β(1)

n d1), (49)

where

β(i)n =

√ε

(i)r k2

0 − (kn + kt)2, (50)

kt =√

ε(0)r k0 sin(θ ), (51)

with kt being the transverse (to z) wave number. For thedefinition of the An coefficients, the same field profiles as in(7) are considered, and therefore their expressions are similarto those in (3) but now incorporate the phase shift imposed bythe incident plane wave:

An =

⎧⎪⎨⎪⎩[J0((kn + kt)w/2)

]2, TM incidence;[

2 J1((kn+kt)w/2)(kn+kt)w/2

]2, TE incidence.

(52)

Regarding the criterion for the number of low-orderharmonics, we now define N as the number of propagativenegative harmonics (n < 0) plus 1 (the first evanescentone) inside the highest-permittivity medium at the highestfrequency of interest, namely,

N =⌈(√

εr,max +√

ε(0)r sin θ

) p

λ0,min

⌉(53)

(for positive angles of incidence the number of propagativepositive harmonics is at most equal to the number of prop-agative negative harmonics). All the harmonics with |n| � N

are thus considered low-order harmonics whose contribution tothe above sums is taken rigorously without any approximation.The harmonics with |n| � N + 1 are all far below the cutoffin the frequency range of interest, and thus we can use thefollowing approximation:

kn + kt ≈ kn, (54)

βn ≈ −jk|n|, (55)

which implies that, once again, their global contribution tothe equivalent circuit can be accounted for by frequency-independent capacitances or inductances. Taking into account(55), the same criterion for M in (28) can be applied here, andonly those high-order harmonics with |n| � M are consideredin the elements that represent coupling through evanescentfields.

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CARLOS MOLERO et al. PHYSICAL REVIEW E 93, 013306 (2016)

FIG. 15. Transmission coefficient (magnitude) versus normalizedfrequency for a four-slit-array stacked structure under oblique TMincidence (θ = 20◦). Parameters: w = 0.1 p, d1 = 0.4 p, d2 = 0.3 p,d3 = 0.2p, ε(0)

r = 1, ε(1)r = 2.2, ε(2)

r = 4, ε(3)r = 3.

As an example of application, Fig. 15 shows the reflectioncoefficient versus frequency for the case of four stackedslit arrays under oblique TM incidence with θ = 20◦. As inprevious comparisons, the agreement with the results fromHFSS is very good.

Finally, it should be mentioned that the range of applicationof the model is more reduced for oblique incidence than fornormal incidence, as expected. As reasonable estimates ofthe validity limits, we have found that the model is valid forfrequencies up to w/λdiel � 0.2 for oblique TM incidence andw/λeff � 0.5 for oblique TE incidence.

V. CONCLUSIONS

A wideband equivalent � circuit has been derived for apair of coupled slit arrays illuminated by TM or TE normallyincident plane waves. Fully analytical expressions are derivedfor the �-circuit elements, which are valid both for slightlyand for tightly coupled gratings. The numerical series definingthe circuit components are physically interpreted in terms ofdistributed (low-order-mode) and lumped (high-order-mode)contributions, in such a way that explicit networks canbe obtained for each specific configuration and maximumoperation frequency. The deduced �-circuit model showsseparate contributions of the internal and external fields to theparallel elements, which allows for a straightforward extensionof the circuit model to the case of stacked structures withan arbitrary number of slit gratings separated by dielectricslabs. Specifically, the equivalent circuit is obtained just bycascading the elementary internal � building blocks. Themodel can also be used for the dispersion analysis of theBloch modes of infinitely long periodic stacks. The extensionof the formulation to oblique incidence is straightforward. Thevalidity of the derived circuit models has been systematicallyverified by proper comparison with numerical data generatedby the MoM, by the commercial simulator HFSS, and also bysome available experimental results.

ACKNOWLEDGMENTS

This work was supported by the Spanish Ministerio deEconomıa y Competitividad with European Union FEDERfunds (project TEC2013-41913-P) and by the Consejerıa deEconomıa, Innovacion, Ciencia y Empleo, Junta de Andalucıa(project P12-TIC-1435).

APPENDIX: TRANSMISSION LINE EQUIVALENTTO A � CIRCUIT

The role of the low-order modes associated with the internal� topology in Sec. II B can be treated as individual n-order� circuits. The n-order modal parallel and series admittancesare given by

Y (1)p, n = jAnY

(1)n tan

(β(i)

n d1/2), (A1)

Ys, n = −jAnY(1)n csc

(β(1)

n d1). (A2)

Using the transmission matrix formulation [62], the ABCD

parameters associated with a � topology are given by

[ABCD]n =

⎡⎢⎣ 1 + Y

(1)p,n

Ys,n

1Ys,n

2Y (1)p,n +

(Y

(1)p,n

)2

Ys,n1 + Y

(1)p,n

Ys,n

⎤⎥⎦. (A3)

Substituting (A1) and (A2) into (A3), it is found after somemanipulations that

[ABCD]n =[Y (1)

n cos(β(1)

n d1)

j An

Y(1)n

sin(β(1)

n d1)

j Y(1)n

Ansin(β(1)

n d1)

Y (1)n cos

(β(1)

n d1)], (A4)

which can be decomposed as the following matrix product:

[ABCD]n =[A

1/2n 00 1

A1/2n

]

×[

Y (1)n cos

(β(1)

n d1)

j 1Y

(1)n

sin(β(1)

n d1)

jY (1)n sin

(β(1)

n d1)

Y (1)n cos

(β(1)

n d1)]

×[

1A

1/2n

0

0 A1/2n

]. (A5)

This cascade of transmission matrices actually representsa transmission-line section of length d1 and characteristicadmittance Y (1)

n terminated by transformers of the turn ratioA

1/2n on both sides [62]. Note that for n = 0, A0 = 1 for either

TE or TM polarization, and thus (A5) reduces to

[Y

(1)0 cos

(β

(1)0 d1

)j 1Y

(1)0

sin(β

(1)0 d1

)jY (1)

0 sin(β

(1)0 d1

)Y

(1)0 cos

(β

(1)0 d1

)], (A6)

which represents a transmission-line section of length d1 andcharacteristic admittance Y

(1)0 .

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WIDEBAND ANALYTICAL EQUIVALENT CIRCUIT FOR . . . PHYSICAL REVIEW E 93, 013306 (2016)

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